MATH-M 344 Chapter Notes - Chapter 10: Boundary Value Problem, Free Variables And Bound Variables, Linear Algebra
Document Summary
M344 section 10. 1 notes- two-point boundary value problems boundary conditions. May be given, especially in physics, differential equations with no initial conditions, but instead. Typical initial value problem- (cid:1877) +(cid:1868)(cid:4666)(cid:4667)(cid:1877) +(cid:1869)(cid:4666)(cid:4667)(cid:1877)=(cid:4666)(cid:4667),(cid:1877)(cid:4666)(cid:2868)(cid:4667)=(cid:2868),(cid:1877) (cid:4666)(cid:2868)(cid:4667)=(cid:1877)(cid:2868) : creates problem for solving (cid:1877) +(cid:1868)(cid:4666)(cid:4667)(cid:1877) +(cid:1869)(cid:4666)(cid:4667)(cid:1877)=(cid:4666)(cid:4667, equation satisfied for some (cid:2009) (cid:1876) (cid:2010); given boundary conditions (cid:1877)(cid:4666)(cid:2009)(cid:4667)=(cid:1877)(cid:2868),(cid:1877)(cid:4666)(cid:2010)(cid:4667)=(cid:1877)(cid:2869, solutions different than those to initial value problems. To solve boundary condition problem: use general solution to equation, use boundary conditions to go from general to specific solution; not necessarily unique (may have. 2 types of boundary value problems solutions (ie. free variables) nonsingular (invertible) matrix: nonhomogeneous- simply does not satisfy homogenous conditions. Possible solutions for matrix equation : none when [|(cid:882)] has row with pivot in augmented column; is singular (non-invertible) matrix, unique solution when no free variables present, matrix system contains vectors in ; Infinitely many solutions when free variable(s) in [|(cid:882)] without pivot in augmented column.
